Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1343 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
| direct product | C22×#1343 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | #99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C22⋊C4 | C24 |
| prime radical | C22×#392 | C2 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C25 ◁ C23×C22⋊C4 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
4122 conjugacy classes of subgroups; only the 322 normal subgroups are shown.
This group has 4122 conjugacy classes of subgroups — too many to draw. Only its 322 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2);(3,4);(5,6)(7,9)(8,11)(10,12)(13,14)(15,19,20,17)(16,18);(6,9)(7,12)(10,11)(14,17)(18,19)92 conjugacy classes — view the full character table.
| maximal subgroup of | #72238064, C2×#129729, C22×#11796, … |
| maximal normal subgroup of | #72238064, C2×#129729, C22×#11796, … |
| maximal quotient of | C2×#129729 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 388641);